CVE 448 Fundamental of Finite Element Method 4 ต.ค.2561.pdf

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CVE 448 Fundamental of Finite Element Method

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1. The bar element shown in Figure 1 is uniform and has nodes 1 and 2. Let the assumed axial displacement field have the form

u =

a₁

+

a2

.x

^3.

a) Determine the shape functions N₁ and N₂ which

u =

NJ]₁x + N₂

d

₂x. (15 marks) Given:

d..x

and

d

2

x

are the nodal axial displacements at node 1 and node 2, respectively.

b) Let

d

1x

=

^{8 and}

d

₂^x

=

^8, determine the axial displacement

u

and strain at

.x =

~ from the

4

shape functions obtained from

u =

a₁

+

a₂

.x

^3. (10 marks) y

tE::::::::::::::::::::L::::::::::::::::::::::.il---~~

x,u

1 2

Figure 1

2.

~-2L-_ _ k_1 _ _ _ 1t--_k_2 _ ___,~

~..., __ ... - - - ~ . . , _ __.,P ____ .... ~

Figure 2

(a) Obtain the global stiffness matrix [K] and equation of the assemblage shown in Figure 2 by using the force/displacement matrix relationships along with the fundamental concepts of nodal equilibrium and compatibility. (15 marks)

(b) Determine the nodal displacement of node 1, the forces in each element, and the reactions.

The axial stiffnesses for the elements 1 and 2 are k1 and k2, respectively. The point load Pis applied at node 1. (20 marks)

(c) Which element has the higher internal force if k₁ > k₂? (5 marks)

2

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3.

3

E,A,L

2

E,A,L

Figure 3

(a) Find the total structure stiffness equations {

F} = [ K]{

d} of the plane truss structure as shown in Figure 3 by using the force/displacement relationships along with fundamental concepts of nodal equilibrium and compatibility. (25 marks)

Let E and A be constant for both elements.

(b) Is this structure stable? Explain or prove by using the finite element concept. (10 marks)

3